The shifted wave equation on Damek-Ricci spaces and on homogeneous trees. (English) Zbl 1275.43010
Picardello, Massimo A. (ed.), Trends in harmonic analysis. Selected papers of the conference on harmonic analysis, Rome, Italy, May 30–June 4, 2011. Berlin: Springer (ISBN 978-88-470-2852-4/hbk; 978-88-470-2853-1/ebook). Springer INdAM Series 3, 1-25 (2013).
Let \(S=N\rtimes A\) be a Damek-Ricci space, where \(N\) is a Heisenberg type group and \(A\cong \mathbb R\). Consider the following shifted wave equation
\[
\partial_{tt} u(x,t)=(\Delta_x+Q^2/4)u(x,t),\quad u(x,0)=f(x),\quad\partial_t |_{t=0} u(x,t)=g(x)
\]
on \(S\). Here \(Q\) denotes the homogeneous dimension of \(N. \) To solve explicitly the wave equation, the authors: (i) prove explicit expressions for the dual Abel transform and its inverse, and (ii) extend Ásgeirsson’s mean value theorem to Damek-Ricci spaces. As an application, they investigate the validity of Huygen’s principle for the shifted wave equation with initial data \((f,g)\) supported in a ball. A similar analysis is carried out in the discrete setting of homogeneous trees. The paper is well written and contains detailed proofs.
For the entire collection see [Zbl 1254.00031].
For the entire collection see [Zbl 1254.00031].
Reviewer: Salem Ben Said (Vandoeuvre-les-Nancy)
MSC:
| 43A85 | Harmonic analysis on homogeneous spaces |
| 35L05 | Wave equation |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 22E30 | Analysis on real and complex Lie groups |
| 22E35 | Analysis on \(p\)-adic Lie groups |
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